According to the Star Trek Encyclopedia (4th ed., vol. 2, p. 328), which confirms the above information, the registry of the Copernicus was "NCC-623". However, a screen cap from the next use of the model as the SS Tsiolkovsky in TNG Remastered's "The Naked Now" clearly showed the registry to be "NCC-640", being illegible in the original airing, and still visible in a couple shots that were not corrected in the episode's remastering.

Michael Okuda remarked years later in this respect, "I seem to recall that Grissom may have been relabeled to serve as another ship (the Copernicus?) in Star Trek III or IV. I didn't try to relabel the model for 'The Naked Now,' partly because we realized that the existing registry would not be legible in standard-def video, but also because we were all so insanely busy at the time that no one could take on an additional project that wasn't likely to be seen on the screen." [1]